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A073608
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a(1) = 1, a(n) = smallest number such that a(n)-a(n-k) is a prime power > 1 for all k.
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0
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OFFSET
| 1,2
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COMMENTS
| Differences |a(i)-a(j)| are prime powers for all i,j. Conjecture: sequence is bounded.
Proof that sequence is complete: Assume there is some k after the term 12. Then {k-1, k-3, k-5} must contain a multiple of 3. Also {k-8,k-10,k-12} also contains a multiple of 3. No prime > 12 is a multiple of 3, so the multiples of 3 are both prime powers. This implies there must be two powers of 3 that have a difference at most 11, but no such pair exists > 12 (only 1,3 and 3,9 qualify.) - Jim Nastos (nastos(AT)gmail.com), Aug 09 2002
There is an elementary proof that no set of seven integers of this kind exists. - Don Reble (djr(AT)nk.ca), Aug 10, 2002.
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EXAMPLE
| a(5) = 10 as 10-8, 10-5, 10-3, 10-1 or 2, 5, 7, 9 are prime powers.
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CROSSREFS
| Cf. A073607.
Sequence in context: A189755 A141436 A189377 * A155945 A096985 A206909
Adjacent sequences: A073605 A073606 A073607 * A073609 A073610 A073611
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KEYWORD
| nonn,fini,full
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AUTHOR
| Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 04 2002
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EXTENSIONS
| Sixth term from Jim Nastos, Aug 09, 2002
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